A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT FOR INTEGRAL HOMOLOGY SPHERES

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1 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT FOR INTEGRAL HOMOLOGY SPHERES KAZUO HABIRO Abstract. We construct an invariant J M of integral homology spheres M with values in a completion Z[] d of the polynomial ring Z[] such that the evaluation at each root of unity ζ gives the the SU(2) Witten-Reshetikhin- Turaev invariant τ ζ (M) of M at ζ. Thus J M unifies all the SU(2) Witten- Reshetikhin-Turaev invariants of M. As a conseuence, τ ζ (M) is an algebraic integer. Moreover, it follows that τ ζ (M) as a function on ζ behaves like an analytic function defined on the set of roots of unity. That is, the τ ζ (M) for all roots of unity are determined by a Taylor expansion at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τ ζ (M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at = Introduction In this paper we construct an invariant of integral homology spheres which unifies the SU(2) Witten-Reshetikhin-Turaev invariants at all roots of unity, which we announced in a previous paper [17] The WRT invariant for integral homology spheres. Witten [77] introduced the notion of Chern-Simons path integral which gives a uantum field theory interpretation of the Jones polynomial [30, 31] and predicts the existence of 3- manifold invariants. Using the uantum group U (sl 2 ) at roots of unity, Reshetikhin and Turaev [73] gave a rigorous construction of 3-manifold invariants, which are believed to coincide with the Chern-Simons path integrals. These invariants are called the Witten-Reshetikhin-Turaev (WRT) invariants. In the present paper, we focus on the WRT invariant for integral homology spheres, i.e., closed 3-manifolds M with trivial first homology groups. If we fix such M, the WRT invariant τ ζ (M) C is defined for each root of unity ζ. (Unlike the case of general closed 3-manifolds, one does not have to specify a fourth root of ζ.) For an integer r 1, τ ζr (M) for ζ r = exp 2π 1 r is also denoted by τ r (M). In the literature, usually τ 1 (M) is not defined, but for our purpose it is convenient to defined it as 1. For integral homology spheres, the version τ r(m) introduced by Kirby and Melvin [38] defined for odd r 3 is eual to τ r (M). Let Z C denote the set of all roots of unity. Define the WRT function of M τ(m): Z C, by τ(m)(ζ) = τ ζ (M). The behavior of the function τ(m) is of interest here. Date: May 12, This research was partially supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B),

2 2 KAZUO HABIRO For τ ζ (M) with ζ roots of unity of odd prime (power) orders, there have been more extensive studies than for the other cases. Murakami [62] proved that if ζ Z is of odd prime order, then τ ζ (M) Z[ζ], hence it is an algebraic integer. Ohtsuki [67] extracted from the τ ζ (M) for ζ of odd prime orders a power series invariant, known as the Ohtsuki series of M τ O (M) = 1 + λ n (M)( 1) n Q[[ 1]]. n=1 Lawrence [44] conjectured, and Rozansky [75] later proved, that τ O (M) Z[[ 1]] and that, for each ζ Z of odd prime power order p e, τ O (M) =ζ converges p- adically to τ ζ (M). In this sense, the Ohtsuki series unifies the WRT invariants at roots of unity of odd prime power orders. (For generalizations of the abovementioned results to rational homology spheres and to invariants associated to other Lie groups, see [49, 50, 59, 60, 63, 68, 76].) The proofs of the above-mentioned results depends heavily on the fact that if ζ Z is of prime power order, then ζ 1 is not a unit in the ring Z[ζ]. Otherwise, ζ 1 is a unit in Z[ζ], and expansions in powers of ζ 1 do not work The ring Ẑ[] of analytic functions on the set of roots of unity. The invariant J M of an integral homology sphere M which we construct in the paper takes values in a completion Ẑ[] of the polynomial ring Z[], which was introduced in [17] and studied in [18]. One of the simplest definitions of Ẑ[] is where we set Ẑ[] = lim n Z[]/(() n ), () n = (1 )(1 2 ) (1 n ). The ring Ẑ[] may be regarded as the ring of analytic functions defined on the set Z of roots of unity. This statement is justified by the following facts. (The following overlaps those in [17, 18].) First of all, each element of Ẑ[] can be evaluated at each root of unity. That is, for each ζ Z, the evaluation map ev ζ : Z[] Z[ζ], f() f(ζ), induces a (surjective) ring homomorphism ev ζ : Ẑ[] Z[ζ], since ev ζ (() n ) = 0 if n ord(ζ). It is often useful to write f(ζ) = ev ζ (f()) for f() Ẑ[]. Second, each element of Ẑ[] can be regarded as a (set-theoretic) function on the set of roots of unity. This means that each f Ẑ[] is determined uniuely by the values ev ζ (f) Z[ζ] for all ζ Z, or, euivalently, the function (1.1) ev Z : Ẑ[] ζ Z Z[ζ], f() (f(ζ)) ζ Z. is injective [18, Theorem 6.3]. Here, in a natural way, ζ Z Z[ζ] can be regarded as a subring of the ring of C-valued functions on Z. Third, each element of Ẑ[] has a power series expansion in ζ for each root of unity, and any such power series determines Ẑ[]. In fact, for each ζ Z, the

3 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT 3 inclusion Z[] Z[ζ][] induces a ring homomorphism (1.2) ι ζ : Ẑ[] Z[ζ][[ ζ]], since, for each i 0, () n is divisible by ( ζ) i in Z[ζ][] if n i ord(ζ). Since ι ζ is injective [18, Theorem 5.2], each f Ẑ[] is determined by the power series ι ζ (f). ι ζ (f) may be regarded as the Taylor expansion of f, see Section Fourth, each element of Ẑ[] is completely determined by its values on a subset Z Z if Z has a limit point. Otherwise, not completely. To explain what this means, we introduce a topology on the set Z, which is different from the usual one induced by the topology of C. Two elements ζ, ξ Z are said to be adjacent if ζξ 1 is of prime power order, or, euivalently, if ζ ξ is not a unit in Z[ζ, ξ]. A subset Z Z is defined to be open if, for each ζ Z, all but finitely many elements adjacent to ζ is contained in Z. In this topology, an element ξ Z is a limit point of a subset Z Z (i.e., (U \ {ξ}) Z for all neighborhood U of ξ) if and only if there are infinitely many ζ Z adjacent to ξ. We have the following. Proposition 1.1 ([18, Theorem 6.3]). If Z Z has a limit point, then the ring homomorphism (1.3) ev Z : Ẑ[] ζ Z Z[ζ], f() (f(ζ)) ζ Z, is injective. If Z Z has no limit point, then ev Z is not injective, i.e., there is a non-zero analytic function f Ẑ[] vanishing on Z, see Proposition The above-explained properties of Ẑ[] are closely related to the integrality of Ẑ[]. In fact, the completion Q[] = lim Q[]/(() n ) does not behave like Ẑ[], see n [18, Section 7.5] A unified WRT invariant J M with values in Ẑ[]. The following is the main result of the present paper, and follows from Theorems 10.2 and Theorem 1.2. There is an invariant J M Ẑ[] of an integral homology sphere M such that for any root of unity ζ we have (1.4) ev ζ (J M ) = τ ζ (M). The properties of the ring Ẑ[] explained in the last subsection implies that the WRT function τ(m) may be regarded as an analytic function defined on Z. Let us describe some corollaries to Theorem 1.2 and properties of the ring Ẑ[]. (Part of the discussion below overlaps those in [17, 18].) An immediate conseuence of Theorem 1.2 is the following generalization of Murakami s integrality result. Corollary 1.3 (Conjectured by Lawrence [44]). For any integral homology sphere M and for ζ Z, we have τ ζ (M) Z[ζ]. Theorem 1.2 immediately implies Galois euivariance of τ ζ (M). have (1.5) τ α(ζ) (M) = α(τ ζ (M)) Namely, we

4 4 KAZUO HABIRO for α Gal(Q ab /Q) and ζ Z. Here Q ab denotes the maximal abelian extension of Q, which is generated over Q by all roots of unity. (Gal(Q ab /Q) can be identified with the automorphism group of the group Z = Q/Z.) It is well known that the Galois euivariance of τ ζ (M) is implied by Reshetikhin and Turaev s definition. Theorem 1.2 reexplains it in an apparent way. Proposition 1.1 implies the following. Theorem 1.4. The invariant J M is determined by the WRT function τ(m). (Thus J M and τ(m) have the same strength in distinguishing two integral homology spheres.) Moreover, both J M and τ(m) are determined by the values of τ ζ (M) for ζ Z, where Z Z is any infinite subset with a limit point in the sense explained in Section 1.2. The invariant J M unifies not only the WRT invariants but also the Ohtsuki series τ O (M). Namely, we have (Theorem 12.6) (1.6) ι 1 (J M ) = τ O (M). Injectivity of ι ζ in (1.2) for each ζ Z implies that the J M and hence τ(m) are determined by the power series expansion ι ζ (J M ) Z[ζ][[ ζ]]. In particular, J M is determined by τ O (M), in view of 1.6. Thus J M and τ O (M) have the same strength in distinguishing integral homology spheres. As a conseuence, the Le-Murakami- Ohtsuki invariant [52] determines J M and τ(m), since it determines τ O (M) (see [70]). For further properties of J M, see Sections 12 and Organization of the paper. In Section 2, we first recall the definition of the uantized enveloping algebra U h = U h (sl 2 ) of the Lie algebra sl 2, which is a h-adically complete Hopf Q[[h]]-algebra, and then introduce Z[, 1 ]-subalgebras U, where Z[, 1 ] is regarded as a subring of Q[[h]] by setting = exp h. The Hopf algebra structure of U h induces a Hopf algebra structure on U. We also introduce a Z[, 1 ]-subalgebra U ev, which is the even part of U with respect to a natural (Z/2Z)-grading of U. We define completions Ũ and Ũ ev of the algebras U and U ev, and also completed tensor products of copies of Ũ and Ũ ev. Ũ is euipped with a complete Hopf algebra structure induced by the Hopf algebra structure of U. In Section 3, we first recall the ribbon Hopf algebra structure for U h, and then the braided Hopf algebra structure U h, canonically defined for U h. The main observation in this section (Theorem 3.1) is that Ũ ev is euipped with a braided Hopf algebra structure inherited from that for U h. In Section 4, we first recall from [20] the notion of bottom tangles. An n- component bottom tangle T is a tangle in a cube consisting of n arc components whose endpoints lie in a line in the bottom suare of the cube in such a way that between the two endpoints of each arc there are no endpoints of other arcs. Then we adapt the universal invariants for bottom tangles associated to the ribbon Hopf algebra to the case of U h. The universal invariant J T of T takes values in the invariant part Inv(U ˆ n ˆ n h ) of the n-fold completed tensor product Uh of U h, where the invariant part is considered with respect to the standard tensor product left U h - module structure defined using the left adjoint action of U h. The main observation in this section (Theorem 4.1) is that if T is an n-component, algebraically-split, 0-framed bottom tangle, then J T is contained in the invariant part K n of the n-fold

5 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT 5 completed tensor product (Ũ ev) n of Ũ ev. (Recall that a link L is algebraically split if the linking number of any two distinct components are zero.) The proof of Theorem 4.1 uses the braided Hopf algebra structure of Ũ ev. For a 0-framed bottom knot (i.e., 1-component bottom tangle) T, the universal invariant J T of T takes values in the center Z(Ũ ev ) of Ũ ev. In view of a result proved in [19], this implies that we can express J T as an infinite sum p 0 a p(t )σ p, where a p (T ) Z[, 1 ], p 0, and where σ p Z(U ev ), p 0, are a certain basis of Z(U ev ) Z(Ũ ev ). Each a p (T ) gives a Z[, 1 ]-valued invariant of T, which may be regarded also as an invariant of the closure cl(t ) of T. In Section 5, we recall the definition of the colored Jones polynomial J L (W 1,..., W n ) of a framed link L = L 1 L n, where each component L i of L is colored by a finite-dimensional, irreducible representation W i of U h (sl 2 ). Then we recall from [20] a formulation of the colored Jones polynomial using universal invariant of bottom tangle: J L (W 1,..., W n ) = (tr W1 tr Wn )(J T ), where T is a bottom tangle whose closure is L, and where tr Wi : U h Q[[h]] is the uantum trace in W i. Recall that for each d 0 there is exactly one (n + 1)-dimensional, irreducible representation of U h up to isomorphism, denoted by V d. We need extensions of the colored Jones polynomials for framed links whose components are colored by linear combinations (over Q(v), Z[v, v 1 ], etc., where v = 1/2 = exp h 2 ) of the V d, which are defined naturally by multilinearity. In Section 6, we relate the universal invariant J T Z(Ũ ev) and the Z[, 1 ]- valued invariants a p (T ), p 0, of a bottom knot T defined in Section 4 to the colored Jones polynomials of the closure cl(t ) of T. Theorem 6.4 identifies a p (T ) with J cl(t ) (P p ), where P p is a Q(v)-linear combination of V 0, V 1,..., V p. In Section 7, we give some remarks on the universal invariant J T Z(Ũ ev ) for a bottom knot. First of all, Z(Ũ ev ) is identified with a completion Λ of Z[, 1, t + t 1 ]. For a knot K = cl(t ) with T a bottom knot, we set J K (t, ) = J T Λ by abuse of notation, which we call the two-variable colored Jones invariant of K. The normalized colored Jones polynomial J K (V n )/J unknot (V n ) Z[, 1 ] is eual to the specialization J K ( n+1, ) of J K (t, ). The specialization J K (1, ) Ẑ[] can be regarded as a universal form for the Kashaev invariants of K. We relate the invariant J K (t, ) to Rozansky s integral version of the Melvin-Morton expansion of the colored Jones polynomials of K, and give several conjectures which generalizes Rozansky s rationality theorem. In Section 8, we consider invariants of algebraically-split links. We define a Z[, 1 ]-algebra P which is spanned by certain normalizations P n of P n, and define a completion ˆP of P. We show that for any n-component, algebraically-split, 0- framed link L and for any elements x 1,..., x n ˆP, there is a well-defined element J L (x 1,..., x m ) Ẑ[], see Corollary 8.3. In the proof, results proved in the previous sections, such as Theorem 4.1, are used. In Section 9, we define an element ω in the ring ˆP. Theorem 9.4 states that for any (m + 1)-component, algebraically-split, 0-framed link L 1 L m K such that K is an unknot, and for x 1,..., x m ˆP, we have J L K (x 1,..., x m, ω 1 ) = J L(K,±1) (x 1,..., x m ).

6 6 KAZUO HABIRO Here L (K,±1) is the framed link in S 3 obtained from L 1 L m by ±1-framed surgery along K. In Section 10, we prove the existence of an invariant J M Ẑ[] of integral homology sphere M (Theorem 10.2). We outline the proof below. Recall that M can be expressed as the result of surgery along an algebraically-split framed link L = L 1 L m in S 3 with framings f 1,..., f m {±1}. Then J M is defined by J M := J L 0(ω f1, ω f2,..., ω fm ), where L 0 is the framed link obtained from L by changing all the framings to 0. By Corollary 8.3, we have J M Ẑ[]. To prove that J M does not depend on the choice of L, we use the twisting property of ω (Theorem 9.4) and a refined version of Kirby s calculus for algebraically-split, ±1-framed links (see Theorem 10.1), which was conjectured by Hoste [28] and proved in [21]. The proof of Theorem 10.2 does not involve any existence proof of the WRT invariants τ ζ (M) at roots of unity ζ, hence can be regarded as a new, unified proof for the existence for τ ζ (M), after establishing the specialization property (1.4). In Section 11, we prove this specialization property (Theorem 11.1). We also give an alternative proof of the existence of J M which uses the existence of τ ζ (M) but does not use Theorem In Section 12, we make several observations and give some applications. In Section 12.1, we observe the behavior of J M under taking connected sums and orientation-reversal. In Section 12.2, we observe the failure of an approach to the conjecture that the WRT invariants τ ζ (M) at any infinitely many roots of unity determine J M. In Section 12.3, we study the power series invariants ι ζ (J M ), including the Ohtsuki series τ O (M) = ι 1 (J M ). In Section 12.4, we give some divisibility results for J M 1, etc., implied by well-known results for τ ζ (M) for ζ of small orders 1, 2, 3, 4, 6, and give some applications of these results to the coefficients of the Ohtsuki series and the power series ι 1 (J M ). We also state a conjecture about the values of the eighth WRT invariant τ 8 (M). In Section 13, we first observe that for any complex number α, there is a formal specialization of J M at = α. Motivated by this observation, for each prime p, we define a p-adic analytic version τ p (M) of the WRT function, which is a p-adic analytic function from the unit circle in the field C p of complex p-adic numbers into the valuation ring of C p. The mod p reduction of τ p (M), denoted by τ mod p (M), is defined on the group of units, F p, in the algebraic closure F p of the field F p of p elements, and takes values in F p. In Section 14, we compute some examples of J M for integral homology spheres obtained as the result of surgery along the Borromean rings A with framings 1/a, 1/b, 1/c with a, b, c Z, and some related knot and link invariants. First we compute the colored Jones polynomials of the Borromean rings. Then we compute the powers of the ribbon element in U h and the powers of the twist element ω. This enables us to compute the invariants of the result of surgery from the Borromean rings by performing surgery along some (possibly all) of the three components by framings in {1/m m Z}. In Section 15, we first generalize the universal invariant J K of a knot in S 3 to knots in integral homology spheres (Theorem 15.3). Using this invariant, we prove that if two integral homology spheres M and M are related by surgery along a knot with framing 1/m with m Z, then J M and J M are congruent modulo

7 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT 7 2m 1 (Theorem 15.6). This result suggests that it would be natural to consider a generalization of Ohtsuki s theory of finite type invariants of integral homology spheres, involving (1/m)-surgeries (m Z) along knots. In Section 16, we give some remarks. In Section 16.1, we discuss the relationships between the unified WRT invariant J M and another approach to unify the WRT invariants by realizing them as limiting values of holomorphic functions on the disk < 1. In Sections 16.2 and 16.3, we mention generalizations of J M to simple Lie algebras and rational homology spheres. In Section 16.4, we announce a generalization of J M to certain cobordisms of surfaces, which includes homology cylinders. 2. The algebra U h (sl 2 ) and its subalgebras In this section, we recall the definition and some properties of the uantized enveloping algebra U h (sl 2 ). Then we define subalgebras U and U ev of U h, as well as their completions Ũ and Ũ ev, which we studied in [19] integers. Let h be an indeterminate, and set v = exp h 2 Q[[h]], = v2 = exp h Q[[h]]. We have Z[, 1 ] Z[v, v 1 ] Q[[h]]. We use two systems of -integer notations. One is the -version : {i} = i 1, {i},n = {i} {i 1} {i n + 1}, {n}! = {n},n, [ i [i] = {i} /{1}, [n]! = [n] [n 1] [1], = {i} n],n /{n}!, for i Z, n 0. These are elements in Z[, 1 ]. (In later sections, we also use () n = ( 1) n {n}!.) The other is the balanced v-version : {i} = v i v i, {i} n = {i}{i 1} {i n + 1}, {n}! = {n} n, [ i [i] = {i}/{1}, [n]! = [n][n 1] [1], = {i} n] n /{n}!, for i Z, n 0. These are elements of Z[, 1 ] vz[, 1 ] Z[v, v 1 ]. These two families of notations are the same up to multiplication by powers of v. The former system is useful in clarifying that formulas are defined over Z[, 1 ]. The latter is sometimes useful in clarifying that formulas have symmetry under conjugation v v The uantized enveloping algebra U h. We define U h = U h (sl 2 ) as the h-adically complete Q[[h]]-algebra, topologically generated by the elements H, E, and F, satisfying the relations where we set HE EH = 2E, HF F H = 2F, EF F E = K K 1 v v 1, K = v H = exp hh 2.

8 8 KAZUO HABIRO The algebra U h has a complete Hopf algebra structure with the comultiplication : U h U h ˆ U h, the counit ɛ: U h Q[[h]] and the antipode S : U h U h defined by (H) = H H, ɛ(h) = 0, S(H) = H, (E) = E 1 + K E, ɛ(e) = 0, S(E) = K 1 E, (F ) = F K F, ɛ(f ) = 0, S(F ) = F K. (Here ˆ denotes the h-adically completed tensor product.) For p Z, let Γ p (U h ) denote the complete Q[[h]]-submodule of U h topologically spanned by the elements F i H j E k with i, j, k 0, k i = p. This gives a topological Z-graded algebra structure for U h U h = ˆ p Z Γ p (U h ). (Here ˆ denotes h-adically completed direct sum.) The elements of Γ p (U h ) are said to be homogeneous of degree p. For a homogeneous element x of U h, the degree of x is denoted by x The subalgebras U and U ev of U h. Set e = (v v 1 )E, F (n) = F n /[n]!, F (n) = F n K n /[n]! = v 1 2 n(n 1) F (n) K n for n 0. Let U denote the Z[, 1 ]-subalgebra of U h generated by K, K 1, e, and F (n) for n 1. The definition of U here is euivalent to that in [19, Section 11]. Let U ev denote the Z[, 1 ]-subalgebra of U generated by K 2, K 2 (n), e, and F for n 1. (U ev is the same as G 0 U in [19].) U is euipped with a (Z/2Z)-graded Z[, 1 ]-algebra structure (2.1) U = U ev (2.2) (2.3) (2.4) KU ev. Later we need the following formulas in U. e m F (n) = Ke = ek, K F (n) = n F (n) K, [ ] m + n F (m) F (n) = mn F m (m+n), min(m,n) p=0 Here, for i Z and p 0, we set where n(m p) [ m p ] F (n p) {H m n + 2p},p e m p. {H + i},p = {H + i} {H + i 1} {H + i p + 1}, {H + j} = H+j 1 = j K 2 1 for j Z. Note that {H + j}, {H + i},p Z[, 1 ][K 2, K 2 ]. The following is a -version of [19, Proposition 3.1], which can be easily proved using (2.2) (2.4).

9 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT 9 Lemma 2.1. U (resp. U ev) is freely spanned over Z[, 1 ] by the elements F (i) K j e k (resp. F (i) K 2j e k ) with i, k 0 and j Z. Recall from [19] that U inherits from U h a Hopf Z[, 1 ]-algebra structure, which can be easily verified using the following formulas, which we also need later. (2.5) (2.6) (2.7) (2.8) (e n ) = (K i ) = K i K i, S ±1 (K i ) = K i, n [ n e j] n j K j e j, ( F n (n) ) = F (n j) K j F (j), j=0 S ±1 (e n ) = ( 1) n 1 2 n(n 1) K n e n, S ±1 ( F (n) ) = ( 1) n 1 2 n(n 1) K n F (n), j=0 ɛ(k i ) = 1, ɛ(e n ) = ɛ( F (n) ) = δ n,0. For n 0, the n-output comultiplication [n] : U h U ˆ n h is defined inductively by [0] = ɛ, and [n+1] = ( [n] id) for n 0. For x U h and n 1, we write [n] (x) = x (1) x (n) Adjoint action. Let : U h ˆ U h U h denote the (left) adjoint action defined by (x y) = x y = x (1) ys(x (2) ) for x, y U h. We regard U h as a left U h -module via the adjoint action. Since we have U U U, we may regard U as a left U -module. For each homogeneous element x U h, we have (2.9) (2.10) (2.11) K i x = i x x for i Z, n [ ] e n x = ( 1) j 1 2 j(j 1)+j x n e j xe j j=0 for n 0, n F (n) x = ( 1) j 1 2 j(j 1)+j x F (n j) x F (j) for n 0. j=0 Proposition 2.2. U ev is a left U -submodule of U. Proof. It follows from (2.9) (2.11) that if x is a homogeneous element of U ev, then we have y x U ev for y = K i, e n, F (n) with i Z, n 0. Since these elements generate U, we have the assertion Filtrations and completions. Here we introduce filtrations for the algebras U and U ev, and also for the tensor powers (U ev ) n of U ev. These filtrations produce the associated completions. The definitions below are euivalent to those in [19]. First, we consider the filtration and the completion for U. For p 0, set F p (U ) = U e p U, the two-sided ideal in U generated by e p. Let Ũ denote the completion in U h of U with respect to the decreasing filtration {F p (U )} p 0, i.e., Ũ is the image of the homomorphism (2.12) lim U /F p (U ) U h p 0

10 10 KAZUO HABIRO induced by U U h. (Conjecturally, (2.12) is injective, see [19, Conjecture 7.2].) Clearly, Ũ is a Z[, 1 ]-subalgebra of U h. Second, we consider U ev. For p 0, set F p (U ev ) = F p (U ) U ev We have a (Z/2Z)-grading F p (U ) = F p (U ev completion in U h of U ev with respect to {F p (U ev lim p 0 = U ev e p U ev. ev ). Let Ũ )} p 0, i.e., the image of ) KF p(u ev U ev /F p(u ev ) U h. denote the Note that Ũ ev is a Z[, 1 ]-subalgebra of Ũ, and Ũ has a (Z/2Z)-graded Z[, 1 ]- algebra structure Ũ = Ũ ev KŨ ev induced by (2.1). Note that the elements of Ũ (resp. Ũ ev ) are the elements of U h that can be expressed as infinite sums Ni i=0 j=1 x i,je i y i,j, where N 0, N 1,... 0, and x i,j, y i,j U (resp. x i,j, y i,j U ev) for i 0, 1 j N i. Now, we define the filtration for (U ev) n, n 1, by F p ((U ev ) n ) = n i=1 (U ev ) (i 1) F p (U ev ) (U ev ) (n i) (U ev ) n. Thus, an element of (U ev) n is in the pth filtration if and only if it is expressed as a sum of terms each having at least one tensor factor in the pth filtration. Define the ) n = Ũ ev (Ũ ev ) n to be the completion in ) n with respect to this filtration, i.e., the image of the homomorphism completed tensor product (Ũ ev U ˆ n ev of (U h lim p 0 For n = 0, it is natural to set Thus we have (U ev F p ((U ev ) 0 ) = F p (Z[, 1 ]) = ) n /F p ((U ev ) n ) U ˆ n h. (Ũ ev ) 0 = Z[, 1 ]. { Z[, 1 ] if p = 0, 0 otherwise. In what follows, we will also need the filtrations and completions of other iterated tensor products of U and U ev, whose definitions should be obvious from the above definitions. For example, U U ev has a filtration defined by Moreover, Ũ Ũ ev F p (U U ev ) = F p (U ) U ev + U F p (U ev ). is defined to be the image of lim F p (U U ev p 0 ) U ˆ 2 h.

11 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT The Hopf algebra structure for Ũ. In this subsection, we show that the Z[, 1 ]-algebra Ũ inherits from U h a complete Hopf algebra structure over Z[, 1 ]. (This fact is observed in [19] as a corollary to the case of a Z[v, v 1 ]-form Ũ = Ũ Z[, 1 ] Z[v, v 1 ]. However, it is convenient to provide a direct proof here.) The Hopf Z[, 1 ]-algebra structure of U induces a complete Hopf algebra structure (with invertible antipode) over Z[, 1 ] of Ũ, since (2.5) (2.8) imply 2 (F p (U )) F p+1 2 (U ), ɛ(u ) Z[, 1 ], ɛ(f 1 (U )) = 0, S ±1 (F p (U )) F p (U ), for p 0. Here p+1 p+1 2 denotes the largest integer smaller than or eual to 2. The structure morphisms : Ũ Ũ 2, ɛ: Ũ Z[, 1 ], S : Ũ Ũ of Ũ are eual to the restrictions of those of U h to Ũ. A conseuence of the above fact is that if f : U ˆ i h U ˆ j h, i, j 0, is a Q[[h]]- module homomorphism obtained from finitely many copies of id Uh, P Uh,U h, µ, η,, ɛ and S ±1 by taking completed tensor products and compositions, then we have f(ũ i ) Ũ j. Here P Uh,U h : U ˆ 2 h U ˆ 2 h is defined by P Uh,U h ( x y) = y x. It follows that the adjoint action : U U U induces a left action : Ũ Ũ Ũ, which is eual to the restriction of : U h ˆ U h U h. By Proposition 2.2, restricts to Thus, Ũ ev : Ũ Ũ ev is a left Ũ-submodule of Ũ. Ũ ev. 3. Braided Hopf algebra structure for Ũ ev In this section, we recall a ribbon Hopf algebra structure for U h and show that the associated braided Hopf algebra structure for U h induces that for Ũ ev Ribbon structure for U h. The Hopf algebra U h has a ribbon Hopf algebra structure as follows. The universal R-matrix and its inverse are given by ( R = D v n(n 1)/2 (v v 1 ) n (3.1) F n E n), [n]! (3.2) where n 0 ( R 1 = ( 1) n v n(n 1)/2 (v v 1 ) n F n E n) D 1, [n]! n 0 D = v 1 2 H H = exp( h ˆ 2 H H) Uh 4. In what follows, we use the following notations. R = α β, R 1 = ᾱ β ( = S(α) β ).

12 12 KAZUO HABIRO The ribbon element and its inverse are given by r = S(α)K 1 β = βks(α), r 1 = αkβ = βk 1 α. The associated grouplike element κ U h defined by satisfies κ = K 1. We also use the following notations. κ = ( S(β)α ) r 1 D = D [1] D [2], D 1 = D[1] D [2]. The following properties of D are freely used in what follows. D[2] D [1] = D, ( 1)(D) = D 13 D 23, (3.3) (3.4) (3.5) (ɛ 1)(D) = 1, (S 1)(D) = D 1, D ±1 (1 x) = (K ± x x)d for homogeneous x U h, where D 13 = D [1] 1 D [2] and D 23 = 1 D [1] D [2] = 1 D. We can easily obtain the following formulas. ( R = D 1 2 (3.6) n(n 1) F (n) K n e n), (3.7) (3.8) n 0 R 1 = D 1( ( 1) n F (n) K n e n), n 0 r = n 0( 1) n F (n) v 1 2 H(H+2) e n, (3.9) r 1 = n n(n 1) F (n) K 2n v 1 2 H(H+2) e n. We have R ±1 D ±1 (Ũ Ũ), and r ±1 v 1 2 H(H+2) Ũ ev Braided Hopf algebra structure for U h. Let Mod Uh denote the category of h-adically complete left U h -modules and continuous left U h -module homomorphisms. The category Mod Uh is euipped with a standard braided category structure, where the braiding ψ V,W : V W W V of two objects V and W in Mod Uh is defined by ψ V,W (v w) = βw αv for v V, w W. The inverse ψ 1 V,W : W V V W of ψ V,W is given by ψ 1 V,W (w v) = S(α)v βw for v V, w W. We regard U h as an object of Mod Uh, euipped with the adjoint action. For simplicity, we write ψ = ψ Uh,U h. Thus the braiding and its inverse for U h satisfy for x, y U h. ψ(x y) = (β y) (α x), ψ 1 (x y) = (S(α) y) (β x)

13 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT 13 Let U h = (U h, µ, η,, ɛ, S) denote the transmutation [56, 57] of U h, which is a standard braided Hopf algebra structure in Mod Uh associated to U h. Here µ: U ˆ 2 h U h, η : Q[[h]] U h, ɛ: U h Q[[h]], are the structure morphisms of U h, and : U h U ˆ 2 h, S : U h U h are the twisted versions of comultiplication and antipode defined by (3.10) (3.11) (x) = x (1) S(β) (α x (2) ), S(x) = β S(α x) for x U h. The inverse S 1 : U h U h of S is given by (3.12) S 1 (x) = S 1 (α x)β Braided Hopf algebra structure for Ũ ev. Unlike Ũ, the even part Ũ ev of Ũ does not inherit a Hopf algebra structure from U h. However, we have the following. Theorem 3.1. The braided Hopf algebra structure of U h induces a braided Hopf algebra structure with invertible antipode for Ũ ev. In other words, for with f : U ˆ i h U ˆ j h and the induced map f : (Ũ ev f {ψ, ψ 1, µ, η,, ɛ, S, S 1 } (i, j 0) we have f((ũ ev ) i ) (Ũ ev ) j, ) i (Ũ ev ) j is continuous. Corollary 3.2. Suppose that f : U ˆ i h U ˆ j h, i, j 0, is a Q[[h]]-module homomorphism obtained from finitely many copies of 1 Uh : U h U h, ψ ±1 : U ˆ 2 h U ˆ 2 ˆ 2 h, µ: Uh U h, η : Q[[h]] U h, : U h U ˆ 2 h, ɛ: U h Q[[h]], S ±1 : U h U h by taking iterated tensor products and compositions. Then we have f((ũ ev ) i ) (Ũ ev ) j. Theorem 3.1 follows immediately from the following. Proposition 3.3. If f {ψ, ψ 1, µ, η, ɛ, S, S 1 } with f : U ˆ i h then we have (3.13) f(f p ((Ũ ev ) i )) F p ((Ũ ev ) j ) for p 0. Moreover, we have (3.14) (F p (Ũ ev ev )) F p+1 ((Ũ ) 2 ) for p 0. 2 U ˆ j h (i, j 0),

14 14 KAZUO HABIRO Proof. The assertion is obvious for f = µ, η, ɛ. The case f = ψ ±1 follows from the following formulas (3.15) ψ(x y) = 1 2 n(n+1)+( x n) y (e n y) ( F (n) x), (3.16) n=0 ψ 1 (x y) = ( 1) n ( x +n) y ( F (n) y) (e n x) n=0 for homogeneous elements x, y U h. Consider the case f = S ±1. Using (3.11) and (3.12), we obtain (3.17) (3.18) S(x) = S 1 (x) = 1 2 n(n+1) e n K x n S( F (n) x), n=0 1 2 n(n 1) n x S 1 ( F (n) x)k x n e n. n=0 Using these formulas, we see easily that S(F p (Ũ ev )) F p (Ũ). Hence, it remains to show that if x, y U ev are homogeneous, then each term in (3.17) and (3.18) is contained in U ev. This follows, since for any homogeneous x U ev we have K x S ±1 (x) U ev by (2.5) and (2.7). Finally, we prove (3.14). By computation, (3.19) (x) = ( 1) n n x (1) K x (2) e n ( F (n) x (2) ), n=0 where (x) = x (1) x (2). Using this formula, we easily see that (F p (Ũ ev )) F p+1 (Ũ Ũ ev ev ). It suffices to show that if x U is homogeneous, then each 2 term in (3.19) is contained in (U ev ) 2. This follows, since we have (3.20) (U ev ) U ev (U ev ) 0 + KU ev (U ev ) 1, where (U ev ) 0 (resp. (U ev ) 1 ) denotes the Z[, 1 ]-submodule of U ev spanned by the homogeneous elements of even (resp. odd) degrees. The inclusion (3.20) can be easily verified using (2.5) and (2.6). Remark 3.4. Ũ also has a braided Hopf algebra structure inherited from that of U h, i.e., Theorem 3.1 holds if we replace Ũ ev with Ũ. Moreover, the (Z/2Z)-grading for Ũ is compatible with the braided Hopf algebra structure: for i, j {0, 1}. (K i Ũ ev ψ ±1 (K i Ũ ev ) Ki Ũ ev K j Ũ ev K i Ũ ev ) K j Ũ ev K i Ũ ev,, S±1 (K i Ũ ev ) Ki Ũ ev, 4. Universal sl 2 invariant of bottom tangles In this section, we recall the definition of the universal invariant of bottom tangles, and prove necessary results.

15 T 1 T 2 T 3 L 1 L 2 L 3 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT 15 (a) (b) Figure 4.1. (a) A 3-component bottom tangle T = T 1 T 2 T 3. (b) Its closure cl(l) = L 1 L 2 L Bottom tangles. Here we recall from [20] the notion of bottom tangles. An n-component bottom tangle T = T 1 T n is a framed tangle in a cube, which is drawn as a diagram in a rectangle as usual, consisting of n arcs T 1,..., T n such that for each i = 1,..., n the component T i runs from the 2ith endpoint on the bottom to the (2i 1)st endpoint on the bottom, where the endpoints are counted from the left. For example, see Figure 4.1 (a). (Here and in what follows, we use the blackboard framing convention.) For each n 0, let BT n denote the set of the isotopy classes of n-component bottom tangles. Set BT = n 0 BT n. The closure cl(t ) of T is the n-component, oriented, ordered framed link in S 3, obtained from T by pasting a -shaped tangle to each component of L, as depicted in Figure 4.1 (b). For any oriented, ordered framed link L, there is a bottom tangle whose closure is isotopic to L. The linking matrix of a bottom tangle T = T 1 T n is defined as that of the closure T. Thus the linking number of T i and T j, i j, is defined as the linking number of the corresponding components in cl(t ), and the framing of T i is defined as the framing of the closure of T i. A link or a bottom tangle is called algebraically-split if the linking matrix is diagonal. For n 0, let BT 0 n denote the subset of BT n consisting of algebraically-split, 0-framed bottom tangles. Set BT 0 = n 0 BT0 n BT Universal sl 2 invariant of bottom tangles. For each ribbon Hopf algebra H, there is a universal invariant of links and tangles from which one can recover the operator invariants, such as the colored Jones polynomials. Such universal invariants has been studied in [42, 43, 72, 53, 66, 33, 35, 34]. Here we need only the case of bottom tangles, which is described in [20]. For T = T 1 T n BT n, we define the universal sl 2 invariant J T U ˆ n h of T as follows. We choose a diagram for T, which is obtained from copies of fundamental tangle, see Figure 4.2, by pasting horizontally and vertically. For each copy of fundamental tangle in the diagram of T, we put elements of U h with the rule described in Figure 4.3. We set J T = J (T1) J (Tn) U ˆ n h, where for each i = 1,..., n, the ith tensorand J (Ti) is defined to be the product of the elements put on the component T i. Here the elements read off along each

16 16 KAZUO HABIRO,,,, Figure 4.2. Fundamental tangles: vertical line, positive and negative crossings, local minimum and local maximum. Here the orientations are arbitrary. 1 S ( α) S ( β) S ( α) S ( β) 1 κ 1 κ 1 Figure 4.3. How to put elements of U h on the strings. For each string in the positive and the negative crossings, S should be replaced with id if the string is oriented downward, and by S otherwise. component are written from right to left. Then J T does not depend on the choice of diagram, and defines an isotopy invariant of bottom tangles Universal sl 2 -invariant of algebraically-split, 0-framed bottom tangles. For any left U h -module W, let Inv(W ) denote the invariant part of W, defined by Inv(W ) = {w W x w = ɛ(x)w x U h }. Recall that we regard U h as a left U h -module via the adjoint action. For n 0, the completed tensor product U ˆ n h is euipped with a left U h -module structure n in the standard way: For x = x 1 x n U ˆ n h and y U h we have In particular, U ˆ 0 h subset X Inv(U ˆ n h For n 0, we set y n x = (y (1) x 1 ) (y (n) x n ). = Q[[h]] is given the trivial left U h -module structure. For any ), we set Inv(X) = Inv(U ˆ n h ) X. K n = Inv((Ũ ev ) n ) (Ũ ev ) n. One can easily see that K n is the Ũ-invariant part of the Ũ-module (Ũ ev ) n, i.e., we have K n = {x (Ũ ev ) n y n x = ɛ(y)x for all y Ũ}. The main result of this subsection is the following. Theorem 4.1. If T BT 0 n, n 0, then we have J T K n. To prove Theorem 4.1, we use the following two results from [20]. Proposition 4.2 (U h case of [20, Proposition 8.2]). For any n-component bottom tangle T BT n, we have J T Inv(U ˆ n h ). Proposition 4.3 (U h case of [20, Corollary 9.15]). Let X n U ˆ n h, n 0, be subsets satisfying the following conditions.

17 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT 17 Figure 4.4. The Borromean tangle B BT 0 3. (1) 1 X 0, 1 X 1, and J B X 3. Here B BT 0 3 is the Borromean tangle depicted in Figure 4.4. (2) If x X l and y X m with l, m 0, then x y X l+m. (3) For p, 0 and f {ψ ±1, µ,, S} with f : U ˆ i h (1 p f 1 )(X p+i+ ) X p+j+. Then, for any T BT 0 n, we have J T X n. U ˆ j h, we have Proof of Theorem 4.1. By Proposition 4.2, it suffices to show that J T (Ũ ev) n. We have only to verify the conditions in Proposition 4.3, where we set X n = (Ũ ev ) n. The condition (2) is obvious. The condition (3) follows from Corollary 3.2. To prove (1), it suffices to prove J B (Ũ ev ) 3. Using Figure 4.4, we obtain J B = ᾱ 3 β 1 α 3 S 2 ( β 1 ) ᾱ 1 β 2 α 1 S 2 ( β 2 ) ᾱ 2 S 2 (β 3 )α 2 β3, where R = α i β i and R 1 = ᾱ i β i for i = 1, 2, 3. Using (3.6) and (3.7), we obtain J B = ( 1) n1+n2+n3 1 2 m1(m1+1) 1 2 m2(m2+1) 1 2 m3(m3+1) m 1,m 2,m 3,n 1,n 2,n 3 0 F (n3) D 3 e m1 D 1 D 3 F (m3) S 2 ( F (n2) D 2 S 2 (e m3 D 3 )D 2 D 1 e n1 ) F (n1) D 1 e m2 D (m2) F D 3 e n3, D i 2 D 1 F (m1) S 2 ( D 2 e n2 ) where D = D i D i and D 1 = D i for i = 1, 2, 3. We slide the tensor factors of the copies of D ±1 using (3.5) so that these copies cancel at the cost of inserting powers of K. Thus we obtain (4.1) J B = m 1,m 2,m 3,n 1,n 2,n 3 0 m3+n3 ( 1) n1+n2+n3 P 3 i=1 ( 1 2 mi(mi+1) ni+mimi+1 2mini 1) F (n3) e m1 F (m3) e n1 K 2m2 F (n1) e m2 F (m1) e n2 K 2m3 F (n2) e m3 F (m2) e n3 K 2m1, where the index i should be considered modulo 3. Each term in (4.1) is in (U ev ) 3. For any p 0, all but finitely many terms in (4.1) involve e r with r p, and therefore are contained in F p ((U ev ) 3 ). Therefore, we have J B (Ũ ev ) Universal sl 2 invariant of bottom knots. By a bottom knot, we mean a 1-component bottom tangle. In what follows, we assume that bottom knots are given 0-framing. Thus the set of bottom knots (up to isotopy) is BT 0 1. By Theorem 4.1, for any bottom knot T BT 0 1, we have (4.2) J T Inv(Ũ ev ) = Z(Ũ ev ).

18 18 KAZUO HABIRO Let us recall from [19] the structure of Z(Ũ ev ev ). (In [19], Ũ G 0 (Ũ).) Set C = (v v 1 ) 2 F E + vk + v 1 K 1 Z(U h ), which is a well-known central element. We have Hence C 2 U ev C = (v v 1 ) F (1) K 1 e + vk + v 1 K 1 vku ev Z(U h ). Z(U h ) = Z(U ev ated by C 2, i.e., we have Z(U ev σ p = is denoted by ). As a Z[, 1 ]-algebra, Z(U ev ) is freely gener- ) = Z[, 1 ][C 2 ]. For p 0, set p (C 2 ( i i )) Z(U ev ), i=1 which is a monic polynomial of degree p in C 2. Therefore, Z(U ev ) = Span Z[, 1 ]{σ p p 0}. As for the center of the completion Ũ ev, we have the following. Theorem 4.4 ([19, Theorem 11.2]). The isomorphism Z(U ev ) = Z[, 1 ][C 2 ] induces an isomorphism Z(Ũ ev ) = lim Z[, 1 ][C 2 ]/(σ p ). p 0 Thus, each element in Z(Ũ ev ) is uniuely expressed as an infinite sum p 0 a pσ p, where a p Z[, 1 ] for p 0. This implies the following. Theorem 4.5. If T is a bottom knot, then J T is uniuely expressed as (4.3) J T = p 0 a p (T )σ p, where a p (T ) Z[, 1 ] for p 0. Note that the a p (T ) are invariants of a bottom knot T. In Section 6, we give a formula which express a p (T ) using the colored Jones polynomials of the closure of T. Remark 4.6. There is an obvious one-to-one correspondence between bottom knots and string knots (i.e., a string link consisting of just one arc component running from the above to the bottom). A bottom knot and the corresponding string knot have the same value of the universal invariant. Therefore, we have the result announced in [17, Theorem 2.1], [19, Theorem 1.2], which is the string knot version of Theorem Colored Jones polynomials In this section, we recall the definition of the colored Jones polynomials of framed links.

19 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT Finite-dimensional representations of U h. By a finite-dimensional representation of U h, we mean a left U h -module which is free of finite rank as a Q[[h]]- module. It is well known that, for each n 0, there is exactly one irreducible finitedimensional representation V n of rank n + 1 up to isomorphism, which corresponds to the (n + 1)-dimensional irreducible representation of sl 2. The structure of V n is as follows. Let v0 n V n denote a highest weight vector of V n, which is characterized by Ev0 n = 0, Hvn 0 = nvn 0 and U hv0 n = V n. It is useful to define the other basis elements by Then the action of U on V n is given by v n i = F (i) v n 0 for i = 1,..., n. (5.1) e m v n K ±1 v n i = v n 2i v n i, (5.2) (5.3) i = {n i + m},m vi m n [ ], = mi i + m m F (m) v n i vi+m, n for i = 0,..., n and m 0, where we understand v n i = 0 unless 0 i n Colored Jones polynomials. Let L = L 1 L m be an m-component, framed, oriented, ordered link, and let W 1,..., W m 0 be finite-dimensional representations of U h. We consider each W i as a color attached to the component L i. Then the colored Jones polynomial J L (W 1,..., W m ) of the colored link (L; W 1,..., W m ) is defined as follows. We choose a diagram of L which is obtained by pasting copies of the fundamental tangles, as in the definition of the universal sl 2 invariant in Section 4.2. To each copy of a fundamental tangle in L, we associate a left U h -module homomorphism. To a vertical line contained in L i, we associate to the left U h -module W i if the line is oriented downward, and the dual Wi of W i if the line is oriented upward. To a positive crossing, we associate a braiding operator ψ W,W : W W W W, x y βy αx, where W (resp. W ) are the left U h -modules associated to the string which connects the upper left corner and the lower right corner (resp. the upper right corner and the lower left corner). To a negative crossing, we associate ψ 1 W,W : W W W W, x y ᾱy βx, where W and W are as above. To the last four tangles in Figure 4.3, we associate the following operators ev W : W W Q[[h]], ev W : W W Q[[h]], coev W : Q[[h]] W W, f x f(x), x f f(κx), 1 i x i x i, coev W : Q[[h]] W W, 1 i x i κ 1 x i, respectively, where W = W i if L i is the component of L containing the fundamental tangle, and where the x i are a basis of W and the x i W are the dual basis.

20 20 KAZUO HABIRO By tensoring and composing the operators, we obtain a left U h -module homomorphism from Q[[h]] to Q[[h]], and we define J L (W 1,..., W m ) to be the trace of this homomorphism. Note that J L alone denotes the universal sl 2 invariant of L. The above definition of J L (W 1,..., W n ) is an abuse of notation, but it should not cause confusion. It is well known that J L p/4 Z[, 1 ] Z[ ±1/4 ], where p Z/4Z depends only on the linking matrix of L and on n 1,..., n m. The invariant is normalized so that J () = 1, and J U (V n ) = [n + 1] for n 0, where U denotes the 0-framed unknot Quantum trace and the colored Jones polynomial. If V is a finitedimensional representation of U h, then the uantum trace tr V (x) in V of an element x U h is defined by ( ) tr V (x) = tr V (ρ V (κx)) = tr V (ρ V (K 1 x)) Q[[h]], where ρ V : U h End(V ) denotes the left action of U h on V, and tr V : End(V ) Q[[h]] denotes the trace in V. We have a continuous left H-module homomorphism tr V : U h Q[[h]]. Let L = L 1 L m be an m-component, ordered, framed oriented link in S 3. Choose T BT m such that cl(t ) is isotopic to L. As explained in [20, Section 1.2] we have (5.4) J L (W 1,..., W m ) = (tr W1 tr W2 for finite dimensional representations W 1,..., W m of U h. tr Wm )(J T ) 5.4. Representation rings. For a commutative ring A with unit, let R A denote the A-algebra R A = Span A {V n n 0}. with the multiplication induced by tensor product. Since each finite-dimensional representation of U h is a direct sum of copies of V n, n 0, we may regard R A as the representation ring of U h over A. By the well-known isomorphism of left U h -modules we have the identity in R A V m V n = V m n V m n +2 V m+n, V m V n = V m n + V m n V m+n. As an A-algebra, R A is freely generated by V 1, i.e., R A = A[V1 ]. Thus we identify R A with A[V 1 ]. For y = n a nv n (a n Q[[h]]) and x U h, we set tr y (x) = n a n tr Vn (x). Thus we have bilinear maps tr ( ): R Z[v,v 1 ] U h Q[[h]], tr ( ): R Q(v) U h Q((h)), where Q((h)) denote the uotient field of Q[[h]].

21 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT 21 Similarly, for a link L we have multilinear maps J L : R Z[v,v 1 ] R Z[v,v 1 ] Z[v 1/2, v 1/2 ], J L : R Q(v) R Q(v) Q(v 1/2 ). If the framing of every component of L is even, then J L above take values in Z[v, v 1 ], Q(v), respectively. 6. Knots In this section, we study the relationships between the universal invariant for a bottom knot T and the colored Jones polynomials for the closure of T The elements P n and P n. For each n 0 set n 1 P n = (V 1 v 2i+1 v 2i 1 ) R Z[v,v 1 ], i=0 P n = P n /{2n + 1} 2n R Q(v). Since P n is a monic polynomial in V 1 of degree n for each n 0, it follows that the P n form a basis of R Z[v,v 1 ]. We have the following base change formulas. Lemma 6.1. For n 0, we have n [ ] P n = ( 1) n i [2i + 2] 2n + 1 (6.1) V [n + i + 2] n i i, i=0 n [ ] n + i + 1 (6.2) V n = P 2i + 1 i. i=0 Remark 6.2. A formula essentially the same as (6.2) formulated using the Kauffman bracket skein module of a solid torus has appeared in [16, Proposition 5.1.], [58, Euation 47]. (The former contained an incorrect sign.) Proof. The proofs of (6.1) and (6.2) are by inductions on n using P n = P n 1 (V 1 (v 2n 1 + v 2n+1 )) (n 1), V n = V 1 V n 1 V n 2 (n 2), respectively. For an alternative proof of (6.2), see Remark The reduced Jones polynomials of knots. We prove the following result in Section 6.3. Proposition 6.3. For m, n 0 we have (6.3) tr P m (σ n ) = δ m,n. Using Proposition 6.3 we obtain the following. Theorem 6.4. For a bottom knot T BT 1 with closure K = cl(t ), we have (6.4) J T = n 0 J K (P n )σ n.

22 22 KAZUO HABIRO Figure 6.1. The bottom tangle c + BT 2. Proof. We express J T as in Theorem 4.5. Then, applying tr P m to (4.3), we have by Proposition 6.3 tr P m (J T ) = a p (T )tr P m (σ p ) = a p (T ). p 0 Therefore, we have (6.4). Theorems 4.5 and 6.4 implies that for a knot K, we have J K (P n ) Z[, 1 ] for n 0. We call J K (P n ) the nth reduced Jones polynomial of K. It is clear that J K (P 0 ) = 1. For n = 1, we have J K (P 1 ) = (J K(V 1 ) [2])/{2}{3}. The uantity 2 J K (P 1 ) is known as the reduced Jones polynomial of K, and appears in the original paper of Jones [31, Proposition 12.5]. Some examples of J K (P n ) are given in Section It is clear that the universal invariant J T determines the colored Jones polynomials J K (V n ), hence the reduced Jones polynomials J K (P n ). Conversely, Theorem 6.4 implies that the universal invariant J T is determined by the J K (P n ), hence by the J K (V n ) as is presumably well known. By (6.2), we have n {n i} 2i+1 (6.5) J K (V n ) = J K (P i ), {1} i=0 where the sum may be replaced with i=0 since {n i} 2i+1 = 0 for i > n. (Conversely, one can use (6.1) to obtain a formula for the reduced Jones polynomials in terms of the colored Jones polynomials.) (6.5) implies the following. Proposition 6.5. If K is a knot, then for each n 0 the nth colored Jones polynomial J K (V n ) is determined modulo ({2n+1} 2n ) by J K (V 0 ), J K (V 1 ),..., J K (V n 1 ) Proof of Proposition The homomorphism ξ : R Q[[h]] Z(U h ). Let c + denote the 2-component bottom tangle depicted in Figure 6.1. We have J c+ = (S 1)(R 21 R) = S(α)S(β ) α β Inv(U ˆ 2 h ), where R = α β = α β. Define a continuous Q[[h]]-algebra homomorphism ξ : R Q[[h]] Z(U h ) by ξ(y) := (1 tr y )(J c+ ) = S(α)S(β )tr y (α β) for y R Q[[h]]. Indeed, we can verify (6.6) ξ(v V ) = ξ(v )ξ(v ) graphically as depicted in Figure 6.2. We can also verify ξ(v 1 ) = C by computation.

23 A UNIFIED WITTEN-RESHETIKHIN-TURAEV INVARIANT 23 ξ (VV )= V V = = V V V V = ξ (V) ξ (V ) Figure 6.2. A graphical proof of (6.6). Since the C i, i 0, are linearly independent in Z(U h ), it follows that ξ is injective The Hopf link pairing, : R Z[v,v 1 ] R Z[v,v 1 ] Z[v, v 1 ]. Define a symmetric bilinear form, : R Q[[h]] R Q[[h]] Q[[h]] by (6.7) x, y := (tr x try )(J c + ) = tr x (ξ(y)) = J H(x, y), for x, y U h, where H = H 1 H 2 = cl(c + ) denotes the 0-framed Hopf link with linking number 1. It is well known (see [73, 38]) that for m, n 0, we have Therefore,, restricts to a bilinear form We also need the induced bilinear form V m, V n = [(m + 1)(n + 1)]., : R Z[v,v 1 ] R Z[v,v 1 ] Z[v, v 1 ]., : R Q(v) R Q(v) Q(v) The elements S n. Let S denote the Z[, 1 ]-subalgebra of R Z[, 1 ] generated by V1. 2 We have S = Span Z[, ]{V 1 2i i 0}. For n 0, set n n S n = (V1 2 (v i + v i ) 2 ) = (V 2 ( i i )) S. i=1 (The elements corresponding to S n in the Kauffman bracket skein module of solid torus have been introduced in [16].) Clearly, we have i=1 (6.8) ξ(s n ) = σ n for n 0. Observe that S n is a monic polynomial in V 2 1 of degree n. Therefore, S is spanned over Z[, 1 ] by the elements S n, n 0. Proposition 6.6. If m, n 0, then we have (6.9) P m, S n = δ m,n {2m + 1} 2m. (As a conseuence, the map, : R Z[v,v 1 ] S Z[v, v 1 ] is nondegenerate.) Proof. For k 0, set V k = V k/[k + 1]. The map, V k : R Z[v,v 1 ] Z[v, v 1 ], x x, V k is a Z[v, v 1 ]-algebra homomorphism.

24 24 KAZUO HABIRO We will prove (6.10) P m, V 2n = [2n + 1]{n + m} 2m. For p 0, we have V 1 v 2p+1 v 2p 1, V 2n = v2n+1 +v 2n 1 v 2p+1 v 2p 1 = {n p}{n+p+1}. Since, V 2n is a Z[v, v 1 ]-algebra homomorphism, we have m 1 P m, V 2n = m 1 V 1 v 2p+1 v 2p 1, V 2n = {n p}{n + p + 1} = {n + m} 2m. p=0 p=0 Therefore, we have (6.10). Similarly, we can prove (6.11) V m, S n = {m + n + 1} 2n+1 /{1}. By (6.10), P m, V 2n = 0 when 0 n < m. Therefore, we have P m, S n = 0 if 0 n < m. By (6.11), V m, S n = 0 when 0 m < n. Hence, we have P m, S n = 0 when 0 m < n. By (6.10), P m, S m = P m, V 2m = [2m + 1]{2m} 2m = {2m + 1} 2m, since S m V 2m is a linear combination of V 2i for i = 0, 1,..., m 1. This completes the proof. Remark 6.7. One can prove (6.2) as follows. Assume V n = n j=0 a n,jp j, where a n,j Z[v, v 1 ] for j = 0,..., n. Applying, S i to the both sides, we obtain V n, S i = n j=0 a n,j P j, S i. By (6.11) and (6.9), we have {n + i + 1} 2i+1 /{1} = a n,i {2i + 1} 2i, hence a n,i = [ n + i + 1 2i + 1 ] Proof of Proposition 6.3. For each n 0 we have ξ(s n ) = σ n. Therefore, by (6.8), (6.7) and (6.9), we have hence the assertion. tr Pm (σ n ) = tr Pm (ξ(s n )) = P m, S n = δ m,n {2m + 1} 2m, 7. Remarks on knot invariants In this section we discuss some conseuences of the results in Section 6. The reader may skip this section in the first reading, since it is not necessary for the proof of the existence of the invariants J M for integral homology spheres. Throughout this section, let T be a bottom knot and let K = cl(t ) be the closure of T. By abuse of notation, we set J K = J T Z(Ũ ev ) The two-variable colored Jones invariant. Part of the following overlaps [17, Section 3.3], where some results are stated without proofs. It follows from the results in the previous sections that we have J K = J T = J K (P n )σ n Z(Ũ ev ) = lim Z[, 1 ][C 2 ]/(σ k ). k n 0 Thus J K may be regarded as an invariant with two variables and C. It is useful to introduce variables t and α satisfying α 2 = C 2 4 = t + t 1 2.